A consistent example of a hereditarily
-Lindelöf
first countable space of size

István Juhász, Lajos Soukup and
Zoltán Szentmiklóssy

Answering a question raised by Anishkievic and Arhangelski, we
show that if
then there is an -closed and
partial order such that, in , there exists a
0-dimensional, , hereditarily
-Lindelöf
, and
first countable space of cardinality
. The
question if there is such a space (even with
``hereditarily'' dropped) in ZFC remains open.